Q.What does the Cartesian product $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ represent geometrically? Verify that the point $(2, -1, 3)$ belongs to it, and explain why $(0, 0)$ does not.

Step 1: $\mathbb{R}\times\mathbb{R}\times\mathbb{R}=\{(x,y,z):x,y,z\in\mathbb{R}\}$ is the set of all ordered triples of real numbers, identified with three-dimensional space (three mutually perpendicular axes through a common origin).

Step 2: $(2,-1,3)$ has three real entries $2,-1,3$, so it fits the defining pattern $(x,y,z)$ with $x=2,y=-1,z=3$; hence $(2,-1,3)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.

Step 3: $(0,0)$ has only two entries, so it is an element of $\mathbb{R}\times\mathbb{R}$ (a pair), not of $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ (which requires exactly three entries) — the two sets consist of objects of a different 'shape' (pairs versus triples) and are not comparable this way.

Recall that $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ contains ordered TRIPLES only; check how many components each given object has before checking membership.

• Assuming $(0,0)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ by loosely treating it as '$(0,0,0)$ with the last $0$ omitted'.
• Describing $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ as the coordinate plane instead of 3-dimensional space.